In the single-hop RIS scenario, four benchmarks are considered: i)
AM (WMMSE-based) method [1]}; ii) BCD (FP-based)
method [1]}; iii) Random Phase where the phase
shift matrices \(\bigl \lbrace \mathbf {\Theta }_{i}\bigr \rbrace \)  are
randomly initialized and \(\lbrace \mathbf {w}_{i}\rbrace \)  is optimized via minorization-maximization
(MM); iv) Without RIS, i.e., no RIS is used and \(\lbrace \mathbf {w}_{i}\rbrace \) 
is optimized via MM. The comparison in terms of achievable WSR is
depicted in Fig. REF  where we have set \(N=100\) 
and \(\sigma ^{2}=1\) , from which we can see that AM, BCD, and the proposed
BMM approach converge to the same WSR as expected, and the benefits
of introducing multiple RIS to improve WSR is obvious. The convergence
behaviors of different approaches with \(d=200m\)  under single-hop
scenario are shown in Fig. REF , from
which we can see that the serial BMM acquires the least number of
iterations to converge.To make it consistant with all the methods (especailly the double-loop
methods) under comparison, “one iteration” is defined after all
the variable blocks are updated once. The convergence speed can be
better validated with the complement of the convergence plot in terms
of CPU time (i.e., Fig. REF ). The convergence speed in terms of CPU time is further presented
in Fig. REF , where the parallel
BMM achieves the lowest time consumption among all cases. Besides,
the influence of the quantization level of the discrete phase is also
investigated. Fig. REF  showcases the approximation
performance of WSR under discrete schemes compared to the continuous
scheme.
